# Electromagnetic surface wave propagation in a metallic wire and the Lambert $W$ function

## TL;DR

This paper revisits Sommerfeld's classical solution for electromagnetic surface wave propagation in metallic wires, analyzing the convergence of his iterative method and comparing it with numerical and Lambert W function solutions.

## Contribution

It provides an elementary convergence analysis of Sommerfeld's iterative solution and compares it with numerical and Lambert W function approaches.

## Key findings

- Convergence of Sommerfeld's iterative method is established.
- Lambert W function offers an alternative solution approach.
- Comparison shows the effectiveness of different solution methods.

## Abstract

We revisit the solution due to Sommerfeld of a problem in classical electrodynamics, namely, that of the propagation of an electromagnetic axially symmetric surface wave (a low-attenuation single TM$_{01}$ mode) in a cylindrical metallic wire, and his iterative method to solve the transcendental equation that appears in the determination of the propagation wave number from the boundary conditions. We present an elementary analysis of the convergence of Sommerfeld's iterative solution of the approximate problem and compare it with both the numerical solution of the exact transcendental equation and the solution of the approximate problem by means of the Lambert $W$ function.